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Further Number Theory The Division Algorithm The division algorithm uses a theorem & method so as to express any given integer a as a bq r , 0 r b , where b 0 , a, b q & r are unique. Examples Find q & r by the division algorithm such that a bq r where ① a 33& b 4 ② a 140& b 6 33 4q r 140 6q r 33 4 8 1 140 6 23 2 q 8, r 1 q 23, r 2 Questions Use the division algorithm to find q & r such that a bq r where 1) a 25& b 7 2) a 9& b 4 3) a 203& b 8 4) a 150& b 6 Solutions 1) a bq r 25 7 q r 25 7 3 4 2) q 3, r 4 a bq r 3) a bq r 4) a bq r 9 4q r 203 8q r 150 6q r 9 4 2 1 203 8 25 3 150 6 25 0 q 2, r 1 q 25, r 3 q 25, r 0 Proof of the Division Algorithm Either a is a multiple of b q OR s.t. a bq & r 0 . a lies between two consecutive multiples of b bq a and a b(q 1), q . bq a b(q 1) 0 a bq bq b bq a bq r 0 a bq b 0r b 0 r b The Euclidean Algorithm The Euclidean Algorithm involves using repeated applications of the division algorithm to find the greatest common divisor of two integers. It is best shown by example. Examples ① Find the greatest common divisor of 146 & 14. 146 14.10 6 14 6.2 2 6 2.3 0 (146,14) (14, 6) {This is alternative notation} (14, 6) (6, 2) (6, 2) (2, 0) (146,14) 2 ② Find (353, 54) & comment on your result. 353 54.6 29 54 29.1 25 29 25.1 4 (353,14) 1 Numbers are said to be coprime. 25 4.6 1 4 1.4 0 Questions 1. Find the gcd of a) 181 & 481 b) 1246 & 242 c) 2488 & 1040 d) 679 & 388 2. Find the gcd of 264, 186 & 368 by a) Finding the gcd of 186 & 264 b) Finding the gcd of solution to a) & 368. Solutions 1. a) 481 181.2 119 181 119.1 62 119 62.1 57 62 57.1 5 b) 1246 242.5 36 242 36.6 26 36 26.1 10 26 10.2 6 57 5.11 2 5 2.2 1 10 6.1 4 2 1.2 0 4 2.2 0 gcd 2 gcd 1 6 4.1 2 c) 2488 1040.2 408 d) 679 388.1 291 388 291.1 97 1040 408.2 224 291 97.3 408 224.1 184 224 184.1 40 gcd 97 184 40.4 24 40 24.1 16 24 16.1 8 16 8.2 0 gcd 8 2. a) 264 186.1 78 186 78.2 30 78 30.2 18 30 18.1 12 18 12.1 6 12 6.2 0 gcd 6 b) 368 6.61 2 6 2.3 0 gcd 2 Using the Euclidean Algorithm The Euclidean Algorithm can be used to express a gcd as a linear combination of two integers. Examples ① Find a, b such that 280a 117b 1 . Perform the Euclidean Algorithm Rearrangement 280 117.2 46 46 280 2.117 117 46.2 25 46 25.1 21 25 21.1 4 25 117 2.46 21 4.5 1 4 1.4 0 gcd 1 Work backwards to achieve linear combination 21 46 1.25 4 25 1.21 1 21 5.4 1 21 5.4 1 21 5.(25 1.21) 1 21 5.21 5.25 1 6.21 5.25 1 6.(46 1.25) 5.25 1 11.25 6.46 1 11.(117 2.46) 6.46 1 11.117 28.46 1 11.117 28.(280 2.117) 1 28.28 67.117 280a 117b 1 when a 28, b 67. (You can check this numerically) ② Calculate (583,318) & express it in the form 583s 318t where s, t calculated. 583 318.1 265 318 265.1 53 and are to be 53 318 1.(583 1.318) 265 583 1.318 53 318 1.265 53 2.318 1.583 265 53.5 0 gcd 53 583s 318t 53 s 1, t 2 ③ 5612 x 540 y 4. Assuming x, y , find their values. 5612 540.10 212 540 212.2 116 212 116.1 96 116 96.1 20 212 5612 10.540 4 20 1.16 116 540 2.212 96 212 1.116 4 20 (96 4.20) 96 20.4 16 20 16.1 4 16 4.4 0 4 5.20 1.96 20 116 1.96 4 5(116 1.96) 1.96 16 96 4.20 4 20 1.16 4 5.116 6.96 4 5.116 6(212 1.116) 4 11.116 6.212 4 11(540 2.212) 6.212 4 11.540 28.212 4 11.540 28(5612 10.540) 4 291.540 28.5612 x 28, y 291. ④ a) Evaluate d (1292,1558). b) Hence express d in the form 1292 x 1558 y where x, y . a) 1558 1292.1 266 1292 266.4 228 266 228.1 38 266 1558 1.1292 228 1292 4.266 38 266 1.228 228 38.6 0 gcd 38 b) 38 266 (1292 4.266) 38 5.(1558 1.1292) 1.1292 38 5.1558 6.1292 1292 x 1558 y 38 when x 6 and y 5. Number Bases Our standard base is base ten. 46 741 10 4 103 10 2 1 100 10 4 10000 6 1000 7 100 4 10 11 This is the decimal system. We can use the division algorithm to convert a number into another base. The most common are: Binary – base two, using only the integers 0 & 1. Octal – base eight, using the integers 0-7. Hexadecimal – base sixteen, using the integers 0-9 & A, B, C, D, E, F. There are others; base three is tenary, base twelve is duodecimal. Binary 101101 is 45. Why? 1.25 0.24 1.23 1.22 0.21 1.20 32 8 4 1 45 We can use the division algorithm to convert 45 into binary. Bear in mind binary is base 2. ① 45 22.2 1 22 11.2 0 11 5.2 1 5 2.2 1 2 1.2 0 Read upwards for binary number 101101 1 0.2 1 Convert 57 into binary. ② 57 28.2 1 28 14.2 0 14 7.2 0 7 3.2 1 3 1.2 1 1 0.2 1 111001 Questions Convert the following numbers into binary numbers a) 131 b) 17 c) 49 d) 100 e) 432. Octal The octal number system work in a similar way. Bear in mind it has a base of 8 and so we will use the numbers 0 to 7. Examples ① 135 16.8 7 ② 613 76.8 5 207 1145 16 2.8 0 2 0.8 2 76 9.8 4 9 1.8 1 1 0.8 1 Questions Find the octal form of the following numbers. a) 439 b) 642 c) 17 d) 200 e) 1964 f) 3024 Hexadecimal Hexadecimal is again similar, using base 16. From remainder 10 onwards we use letters: A=10 B=11 C=12 D=13 E=14 F=15. Examples Find the hexadecimal form of ② 974 60.16 14 E 60 3.16 12 C 12 0.16 3 ① 2167 135.16 7 135 8.16 7 8 0.16 8 877 3CE Questions Find the hexadecimal numbers equivalent to; a) 6015 b) 579 c) 3114 d) 8163 e) 1001 f) 369 Proof All of the proof content has been covered in Unit 2. We will consider further examples in class. Past Paper Questions 2001 Use the Euclidean algorithm to find integers x and y such that 149 x 139 y 1. (4 marks) 2004 Use the Euclidean algorithm to show that (231,17) 1 where (a, b) denotes the highest common factor of a and b. Hence find integers x and y such that 231x 17 y 1. (4 marks) 2007 Use the Euclidean algorithm to find integers p & q such that 599 p 53q 1. (4 marks) 2012 Use the division algorithm to express 123410 in base 7. (3 marks) 2013 Use the Euclidean algorithm to obtain the greatest common divisor of 1204 and 833, expressing it in the form 1204a 833b, where a and b are integers. (4 marks) 2015 Use the Euclidean algorithm to find integers p and q such that 3066 p 713q 1 . (4 marks)